Fair Division / Apportionment

Fair Division / Apportionment
Fall, 2017

Worksheet 1 #1 Alisha and Balavan are dividing a cake by the cut and choose method (also known as the divider-chooser algorithm). After the cake is cut, Alisha thinks the values of the two pieces are 60% and 40%. Balavan thinks the values are 50% and 50%. Who did the cutting? How do you know?

Worksheet 1 #2 Joan, Henry and Sam are heirs to an estate that includes a vacant lot, a boat, a computer, a stereo and $11,000 in cash. Each heir submits bids summarized in this table. For each heir, find their fair share, the items received, the amount of cash and the final settlement. Joan Henry Sam Vacant Lot 8000 7500 6200 Boat 6500 5700 6700 Computer 1340 1500 1400 Stereo 800 1100 1000 Cash 11,000 Total 27,640 26,800 26,300 Fair Share $9,213 $8,933 $8,767

Joan Henry Sam Cash FS 9,213 8,933 8,767 11,000 - 6333 3454 - 2067
RCD 8,000 2, , Diff 1,213 6, , - 6333 3454 - 2067 leaving $1387 1387 ÷ 3 = $462 to each person

Final Settlements Joan: Vacant Lot = $ 9,675 Henry: Computer + Stereo = $ 9,395 Sam: Boat = $ 9,229

Worksheet 1 #3 Three students are invited to share a cake, on the stipulation that they each receive their fair share as determined by the lone divider algorithm. The students draw straws to find out who becomes the divider, D. The other two become choosers with the first chooser deciding the cake is cut into three pieces (a, b, and c) that represent 45%, 35% and 20% of the cake. The second chooser thinks the pieces are closer to 35%, 35% and 30%. Of the three pieces chooser one finds piece “a” acceptable and chooser “b” thinks pieces “a and b” are acceptable, determine who gets each of the three pieces and tell why.

Estate Division Worksheet #1
The famous actor JJ Hugo recently passed away and left all of his estate to his three sons, MM, PP and QQ. They also found that he has hidden $108,000 in cash under his mattress. The chart below details what his three heirs value the items in his estate. Find the final settlement for each. House Swag ’15 JAG Threads MM $225,000 $79,000 $87,000 PP $219,000 $81,500 $78,500 $12,000 QQ $217,500 $75,000 $68,500 $15,000

Continued MM PP QQ Total bids and cash 499,000 484,000 Fair share 166,333 161,333 Items received House Jag Swag Threads Value of item received 312,000 81,500 15,000 Initial cash -145,667 84,833 146,333 Share of remaining cash 7500 Final settlement

Final Settlement Statements
MM— House + Jag – 145, = $173,833 PP— Swag + 84, = $173,833 QQ— Threads + 146, = $168,833

Anne, Beth and Jay are heirs to an estate that includes a computer, a used car, and a stereo. Each heir has submitted bids for the items in the estate as summarized in the following table. Find the final settlement for each heir after applying the estate division algorithm you learned in this unit. Anne Beth Jay Computer 1800 1500 1650 Car 2600 2400 2000 Stereo 1000 800 1200

Continued Total bids and cash Fair share Items received
Anne Beth Jay Total bids and cash 5400 4700 4850 Fair share 1800 1567 1617 Items received Computer Car Stereo Value of item received 4400 1200 Initial cash received -2600 417 Share of remaining cash 205 Final settlement $2005 $1772 $1822

Akeem, Budro, Chiara, and Diamond are heirs to an estate that includes a mansion in Malibu, a hundred foot yacht, a brand new rolls royce, and a collection of jewelry. In addition, the estate has cash assets of $9 million. If the chart below shows how much the four heirs value each item, calculate the final settlement for each heir. Mansion Yacht Rolls Royce Jewelry Akeem $5,600,000 $3,000,000 $120,000 $65,000 Budro $5,450,000 $2,900,000 $119,500 $58,900 Chiara $5,660,000 $117,000 $55,000 Diamond $5,650,000 $3,150,000 $116,000 $56,000

Share of remaining cash
Akeem Budro Chiara Diamond Total bids and cash 17,785,000 17,528,400 17,832,000 17,972,000 Fair share 4,446,250 4,382,100 4,458,000 4,493,000 Items received Rolls Jewelry -- Mansion Yacht Value of item received 185,000 5,660,000 3,150,000 Initial cash received 4,261,250 -1,202,000 1,343,000 Share of remaining cash 53,913 Final settlement $4,500,163 $4,436,013 $4,511,913 $4,546,913

Four heirs, Aaayyy, Beeeee, Seeeee, and Deeeee, are going to inherit an estate with a house, car, $4500 in cash, and jewelry from their aunt. The table below shows their secret bids representing how much they value each item. Find the final settlement for each heir. House Car Jewelry Aaayyyy 54,000 2,900 2,600 Beeeee 49,000 3,100 2,200 Seeeee 55,000 2,850 2,550 Deeeee 53,500 3,300

Continued Aaayyyy Beeeee Seeeee Deeeee Total bids and cash 64,000 58,800 64,900 64,400 Fair share 16,000 14,700 16,225 16,100 Items received House Car Jewelry Value of item received 55,000 6,400 Initial cash received -38,775 9,700 Share of remaining cash 958

Final Settlement Statements
Aaayyyy— 16, = $16,958 Beeeee— 14, = $15,658 Seeeee— House – 38, = $17,183 Deeeee— Car + Jewelry + 9, = $17,058

Sharif, Ahkbar, Hakeem and Tamir recently lost their great uncle who’s estate contained a very nice home, a Rolls Royce automobile, a small restaurant and exactly $380,000 in cash and life insurance. Below you will see what each heir has bid on these items. Find each final settlement. Sharif Ahkbar Hakeem Tamir Home 295,000 310,000 325,000 190,000 Car 89,000 96,000 88,500 92,000 Restaurant 1,200,000 1,220,000 1,350,000

Sharif— 491,000 + 39,281 = Total Value of $530,281
S A H T Fair Share 491, , , ,000 Items Car Home Restaurant Value , , ,350,000 Initial Cash 491, , , ,000 Cash– 380, ,000 = 1,227,000 – 491,000 = 736,000 – 405,500 = 331,000 – 173,375 = 157,125 left over ÷ 4 = $39,281 each Final Settlements: Sharif— 491, ,281 = Total Value of $530,281 Ahkbar— Car + 405, ,281 = Total Value of $540,781 Hakeem— Home + 173, ,281 = Total Value of $537,656 Tamir— Restaurant – 847, ,281 = Total Value of $542,281

Lone Chooser Method Step 1. First Division--The two dividers split the pizza by the divider-chooser method. Step 2. Second Division--Each divider now divides his part into three parts he considers equal. Step 3. Selection--The chooser picks one piece from each divider, and each divider keeps whatever he has left. Again, we have fair shares for everybody.

Fair Division Summary Summing up what we know about Fair Division so far: To be considered fair, each affected party should be part of the solution. Fair is not necessarily equal. Value is part of fair division problems, both intrinsic value and extrinsic value. Everybody is guaranteed their “fair share” Divider-Chooser Method; Lone Divider Method; Lone Chooser Method Estate Division Algorithm

Warm-Up Problem Anne, Beth and Jay are heirs to an estate that includes a computer, a used car, and a stereo. Each heir has submitted bids for the items in the estate as summarized in the following table. Find the final settlement for each heir after applying the estate division algorithm you learned in this unit. Anne Beth Jay Computer 1800 1500 1650 Car 2600 2400 2000 Stereo 1000 800 1200

Remember This Scenario
The sophomore, junior and senior classes of Cox Mill High School have 398, 337 and 319 member respectively. If the student council is composed of 20 members divided among each of the three classes, determine a fair number of seats on the council for each class…and explain why you chose that approach…and why you think it is fair. Turn and talk to your neighbor and try to come up with a “fair” way to distribute the seats based on some mathematical approach.

Apportionment

Apportionment Defined
Wikipedia defines apportionment as (derived from Latin portio, share) meaning distribution or allotment in proper shares. In economics it is the division and sharing carried out according to a plan or formula. In accounting it is the division of income and expenses in certain proportion and, in contrast to allocation, over two or more accounts, departments, or entities. In the law, it is the division and distribution of assets and/or liabilities in proportion to the rights and interests of the parties involved.

So what are the common themes?

Fair Division Mom has 50 pieces of candy that she is going to distribute to her five children. How would she do this?

Big Difference!!! Mom has 50 pieces of identical candy and the same five children BUT she decides to distribute the candy in proportion to the number of minutes of chores each child does this week….. So how much candy does each receive? Take a few minutes with a partner to figure this out on your own…..find a method that works and be ready to explain why.

Mom’s Idea of “FAIR” See if you can figure out how much candy each child receives…..
Kareem Brooke Sammy Don Tyra Minutes of Chores 120 85 50 170 135

Historical Perspective Discussion
Consider the original 13 colonies……discussing how they should be represented….what kinds of arguments do you think they had? Do you think they agreed on everything?

Two Houses of Congress Senate Equal representation
House of Representatives Article 1, Section 2 of the Constitution says in part…. “shall be apportioned … to their respective numbers”??????? Unfortunately they didn’t say how that was supposed to be accomplished.

So Where Might You See Apportionment in Action?
Write down three places where you might see apportionment in action…things, people, money, etc divided in a proportional way using a defined process Select your best idea with someone in the room. Take their idea and share it with someone else. Now, let’s talk about the ideas….

So Where Else Might You See Apportionment in Action?
How about people assigned to a shift in a factory? Nurses in a hospital assigned to shifts? Teachers assigned to teach in a high school? Money distributed in a budget? Buses on different routes in a city or a school district. Airplanes to flight routes based on passenger numbers.

Apportionment So now you are beginning to see why the method of apportionment was so concerning to the founding fathers……it begs to question, why not just put a formula in place when the Constitution was written and that article was agreed upon????? Article 1, Section 2 of the Constitution says in part…. “shall be apportioned … to their respective numbers”???????

The Hamilton Method The Hamilton Method, one of several methods of apportionment we study, is named after Alexander Hamilton. It was first used to decide the initial apportionment of the seats in the House of Representatives in That apportionment was vetoed by George Washington and the House was reapportioned that year according to the Jefferson Method. The Hamilton Method did come back into use in 1850 and was then used until 1900. To explain this and other methods, we must define certain terms…

The Hamilton Method Suppose the total population of all states is p and that are there are h seats in the house (we call h the house size). That is, house size is the total number of seats available. We define the standard divisor, s, as follows: or

The Hamilton Method Thus, each state could have a different quota.
Next, we define the term quota as follows: or Thus, each state could have a different quota. In the case of apportioning seats in the House of Representatives, the quota is the number of seats that a state would get if they could have a fractional part of a seat. That is, it is a fractional part of the whole before being rounded to an integer.

Terms and Definitions Lower Quota is the Standard quota rounded down.
Upper Quota is the Standard quota rounded up. Quota Rule is “A state’s apportionment should either be it’s upper or lower quota.”

The Hamilton Method Summarized
The Hamilton Method of apportionment is as follows: Calculate each state’s quota. Temporarily assign each state it’s lower quota. Starting with state’s having the largest fractional part in their original quota, distribute any remaining seats, in order from largest to smallest, until all remaining seats are distributed. The Hamilton Method does not specify what to do in the case of a tie – if two states had the same fractional part in their quota - but this is unlikely to occur.

Forming Our Own Country
Our new country is only made of eight provinces….. Each province has a certain population. Our new country is going to have a ruling council with 50 representatives What is the total population? What is the standard divisor? What is each province’s standard quota?

Population Assignments
Province ,000 Province ,500 Province ,000 Province ,000 Province ,000 Province ,000 Province ,000 Province ,000

Example The country of Parador has six states: Azucar, Bahia, Café, Diamonte, Esmeralda, Felicidad. Their populations are respectively, 1,646,000; 6,936,000; 154,000; 2,091,000; 685,000; 988,000 Parador has Congress with 250 seats…distributed through apportionment… So Find: Each state’s lower quota and upper quota….

State Azucar Bahia Total
Café Diamante Esmeralda Felicidad Total Population 1,646,000 6,936,000 154,000 2,091,000 685,000 988,000 12,500,000 Standard Quota 32.92 138.72 3.08 41.82 13.70 19.76 Lower Quota Upper Quota 32 33 138 139 3 4 41 42 13 14 19 20 246 252 Final Apportionment Republic of Parador: Population Data by State

Warm-Up Exercise Southwest Guilford High School has sophomore, junior, and senior classes of 464, 240, and 196 respectively. The 31 seats on the school's student council are divided among the classes according to the population. How many should each class have if the administration uses the Hamilton Method of Apportionment?

Hamilton’s Method of Apportionment
Step 1: Calculate each state’s Standard Quota… Step 2: Give to each state its lower quota…. Step 3: Give the surplus seats (one at a time) to the states with the largest residues (or fractional parts) until there are no more surplus seats.

Apportionment Approaches Literacy Activity Due Tuesday
Select one of the people below and create a “foldable” that explains their role in apportionment of the seats in the House of Representatives. Your foldable must include: The time frame in which they developed their apportionment method. The reasons their method was needed. The mathematical process of their method. An explanation of how their method is “different” than the other methods. The time frame(s) in which their method has been used in our country. Any other information you feel will help other students understand their method. Thomas Jefferson Daniel Webster Joseph Hill

Hamilton’s Method of Apportionment
Approved by the Congress in 1781 Was vetoed by President Washington First presidential veto Was subsequently adopted and used from 1852 through 1911 when it was replaced by the Webster Method.

Vocabulary Important to Hamilton’s Apportionment
Standard Divisor = Total Population divided by the total number of seats Standard Quota = Individual Population divided by the standard divisor Lower quota, round down….Upper quota, round up Assign the lower quota to each… Extra seats go one by one to the fractional part not represented… So how many seats does each get? quot

Apportionment Approaches Literacy Activity Due Tuesday
Select one of the people below and create a “foldable” that explains their role in apportionment of the seats in the House of Representatives. Your foldable must include: The time frame in which they developed their apportionment method. The reasons their method was needed. The mathematical process of their method. An explanation of how their method is “different” than the other methods. The time frame(s) in which their method has been used in our country. Any other information you feel will help other students understand their method. Thomas Jefferson Daniel Webster Joseph Hill

A Little History The first Congressional bill ever to be vetoed by the President of the United States was a bill in containing a new apportionment of the House based on Hamilton’s Method. The reason for the veto may have been related to the following requirement stated in the U.S. Constitution (Article 1, section 2): The number of people per single seat in the House should be at least 30,000. Remember that the standard divisor represents the average number of people per seat in the nation as a whole. In 1790, there were 15 states and 105 seats in the House. According to the 1790 census, the U.S. population was 3,615,920. Thus the standard divisor would have been 3,615,920/105 = 34,437. So why the veto?

A Little History It is likely that at least one reason George Washington vetoed the 1790 House apportionment as calculated by Alexander Hamilton is that under Hamilton’s apportionment, two seats were assigned to Delaware while the 1790 census indicated a population of 55,540 for Delaware. Therefore, there were actually 55,540/2 = 27,770 people per seat for Delaware, a violation of the Constitution. How did this happen? The answer comes from the fact that Hamilton’s method had awarded Delaware the extra seat by rounding up the quota for Delaware. That is, in 1790, Delaware’s quota was q = (state population)/(standard divisor) = (55,540)/(34,437) = Following Hamilton’s Method and rounding quota’s, it so happened that Delaware was rounded up to get 2 seats.

A Little History Congress was unable to override the Presidential veto, and to avoid a stalemate, they turned to Thomas Jefferson, who had devised another means of apportionment … Jefferson’s Method was the method actually used for the first apportionment of the House, which was finally done in This method was then used until 1840 when Hamilton’s Method made a return. To use Jefferson’s Method we must find a modified divisor such that when dividing each state’s population by this divisor, and rounding down, the sum of the adjusted quotas (seats) is equal to the total number of seats. One justification for this method is that it may seem more fair in the sense that every state’s quota will be rounded down – based on division by the same modified divisor.

Jefferson’s Method – Example #1
Let’s consider a simple example. Suppose there are only three states: Texas, Alabama and Illinois and 100 seats in the House. Suppose the populations are as given in the table. We begin by calculating the standard divisor, which is 20,000/100 = 200. Then we determine the lower quota for each state using the standard divisor… Population Texas 10,030 Illinois 9,030 Alabama 940 Total 20,000

Now we compute the initial apportionment, as defined previously, using the standard divisor of 20,000/100 = 200. Population Lower quotas, using standard divisor of 200 Texas 10,030 50 Illinois 9,030 45 Alabama 940 4 Total 20,000 99 (1 seat still available) Note that not all of the 100 seats have been apportioned

We haven’t apportioned all of the seats using the standard divisor of 200, so we will choose another modified divisor… We choose a modified divisor so that when we round all of the quotas down, they add to the required total. Population Lower quotas (with standard divisor) Texas 10,030 50 Illinois 9,030 45 Alabama 940 4 Total 20,000 99 (so 1 seat still available)

We find the sum of the lower quotas is less than the required total of 100 seats. Therefore, we seek a modified divisor to replace the standard divisor. By trial and error (or as described below) we find that a modified divisor of d = will work. Is this ok? Is it ok to change the divisor? The answer is yes – there is no constitutional (or mathematical) requirement for using the standard divisor. In fact, we are determining a value representing the lowest ratio of people per seat for any of the states. In this case, it can be found by adding one to each state’s apportionment and dividing into the population and then comparing each of these resulting values and taking the largest, which is Population Lower quotas (with standard divisor) Texas 10,030 50 Illinois 9,030 45 Alabama 940 4 Total 20,000 99 (so 1 seat still available)

We find the sum of the lower quotas is less than the required total of 100 seats. Therefore, we seek a modified divisor to replace the standard divisor. By trial and error (or as described below) we find that a modified divisor of d = will work. Population Lower quotas (with standard divisor) Texas 10,030 50 Illinois 9,030 45 Alabama 940 4 Total 20,000 99 (so 1 seat still available) By finding the value of we are essentially solving a problem of optimization – to find the maximize value of the minimum number of people per seat in any state. This was Jefferson’s problem – he wanted to make sure to satisfy the Constitutional requirement that there were at least 30,000 people per seat in the apportionment of the House. So he was looking at the resulting ratios of people per seat in every state and sought to maximize this value – to make sure it was always more than 30,000.

Now, using the modified divisor, we calculator lower modified quotas for each state… Population Initial lower quotas Lower quotas (with modified divisor of 196.6) Texas 10,030 50 Illinois 9,030 45 Alabama 940 4 Total 20,000 99 (so 1 seat still available) Total apportionment = 100

Now, using the modified divisor, we calculator lower modified quotas for each state… Population Initial lower quotas Lower quota using modified divisor of 196.6 Final apportionment Texas 10,030 50 51 Illinois 9,030 45 Alabama 940 4 Total 20,000 99 (1 seat still available) 100

Practice Problem The first grade reading teacher is using stickers to give her students incentive to write more words in their writing journals. She only has 50 stickers and the number of words each of her students wrote is in the table below. Use this information to find the number of stickers each child should receive using the Jefferson Method. Child Jamie Jody Danny Jimbo Sammy Sharon Words 75 89 65 110 98 54

Let’s consider another example of Jefferson’s Method. Suppose a country consists of six different states: A, B, C, D, E, F. Suppose the populations of these states are different and the country has a House of Representatives with 250 seats. How should these 250 seats be apportioned? Let’s use Jefferson’s Method to answer that question. Here are the population figures from that country’s most recent census: A B C D E F total population 1,646 6,936 154 2,091 685 988 12,500 To use Jefferson’s Method, we first need s, the standard divisor. In this case it’s s = 12,500/250 = 50. (That is, on average, throughout the country, there should be about 50 people per Congressional district – or equivalently, about 50 people per Congressional seat.)

population initial app. (divide and round down) A 1646 1646/50 = 32 B 6936 6936/50 =138 C 154 154/50 = 3 D 2091 2091/50 = 41 E 685 685/50 = 13 F 988 988/50 = 19 total 12,500 246 (so 4 seats remain) Initial apportionments are found by dividing population for each state by the standard divisor and rounding down.

population initial app. (divide and round down) A 1646 1646/50 = 32 B 6936 6936/50 =138 C 154 154/50 = 3 D 2091 2091/50 = 41 E 685 685/50 = 13 F 988 988/50 = 19 total 12,500 246 (so 4 seats remain) Initial apportionments are found by dividing population for each state by the standard divisor and rounding down. Because there are still 4 seats to be assigned we will need to increase the apportionments of some of the states… We do this by searching for a new modified divisor…

To find a modified divisor, we seek to maximize the minimum number of people per seat in any given state. Unfortunately, this example is complicated by the fact that we must add a total of 4 seats to reach the required total of 250. We could find the necessary modified divisor by experimentation or by the following procedure… For each of the four remaining seats, we can temporarily add 1 to all of the apportionments – divide that result into the population – and determine the largest ratio. The state with the largest ratio will get the next seat. We then continue that process for each of the remaining seats. population initial apportionment (after rounding down) Standard quotas A 1646 1646/49.6 = B 6936 6936/49.6 = C 154 154/49.6 = D 2091 2091/49.6 = E 685 685/49.6 = F 988 988/49.6 = total 12,500

By experimentation, we find a modified divisor of 49.5 will work. The final result is in the last column and represents the final apportionment… population initial apportionment (after rounding down) Modified quotas Modified lower quotas Final apportionment A 1646 1646/50 = 32 33.25 33 B 6936 6936/50 =138 140.12 140 C 154 154/50 = 3 3.11 3 D 2091 2091/50 = 41 42.24 42 E 685 685/50 = 13 13.84 13 F 988 988/50 = 19 19.95 19 Total 12,500 246 (so 4 seats remain) 252.5 250

Jefferson Method of Apportionment Video
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Worksheet #1 Samantha Jennie Donna Roger Danny Total 254,300 260,100
253,050 252,700 260,000 Fair Share 50,860 52,020 50,610 50,540 52,000 Items Cabin Jewelry Figurines House Car Value of items 26,000 6,500 9,950 172,700 Initial Cash 24,860 44,110 40,590 -120,700 Extra Cash Share 1424

Worksheet #2 Step 1: Find the Total Total is 6,335,000
Step 2: Find the Standard Divisor Standard Divisor = Total Population # of Seats (or pieces) /545 = 11,623 (people per seat)

Worksheet #2 continued Step 3: Set up a Chart and Find the Standard Quotas by dividing individual populations by the standard divisor: Standard Quotas Lower Quotas Final Apportionment Andromedia 68.82 68 69 Pandora 84.31 84 Cyrious 103.23 103 Xenatia 67.53 67 Rabithia 25.38 25 Finanthia 40.86 40 41 Savagia 154.85 154 155 Total XXXXXX 541 (+4)

Worksheet Tonight on the Website

Jefferson Method of Apportionment
Find the standard divisor - This is found using the following formula. Standard divisor = total population total number of seats 2. Find the Standard Quota. This is found using the following formula. Standard Quota = Number of People in Group Standard Divisor

Jefferson Method of Apportionment
3.Truncate each group's data and assign that number of seats (aka find the lower quota). If the number of seats assigned matches the total number of seats to be apportioned, then stop. If not.. what do we do? We find a new divisor…..we MODIFY the Standard Divisor

Worksheet #1 A company is apportioning 175 newly- trained workers among its four manufacturing plants, according to how many units are produced at each plant per day. The output of the four plants is given below. Plant Location: Albuquerque, Boston, Chicago and Denver Units per Day: 1049; 517; 3001;1558 respectively 1) Apportion the workers using Hamilton’s Method. 2) Apportion the workers using Jefferson’s Method.

Hamilton Solution Population Standard Quota (35) Lower Quota
Final Workers Albuquerque 1049 29.97 29 30 Boston 517 14.77 14 15 Chicago 3001 85.74 85 86 Denver 1558 44.51 44 total 6125 172 175

Jefferson Solution Population Lower Quota Modified Quota using 34.8
Albuquerque 1049 29 30.14 –- 30 30.23 – 30 Boston 517 14 14.86 –-14 14.90 – 14 Chicago 3001 86 86.48 – 86 Denver 1558 44 44.90 – 44 6125 173 174 So what now?

Worksheet #2 Five counties of a state can divide its 50 legislative seats in many ways. Determine the number of representatives for each of the five counties using the Hamilton and Jefferson methods. The following table gives the populations of the different counties. County/ Population: Brooke / 48,859; Hopkins / 161,135; Isaac / 87,194; Haley / 596,270; and Wallace /110,006…….. Total Population = 1,003,464 Standard Divisor = /50 = 20,069

Practice Solution--Hamilton
Standard Quotas Lower Quotas Hamilton Final App. Brooke 2.43 2 Hopkins 8.03 8 Isaac 4.34 4 Haley 29.71 29 30 Wallace 5.48 5 6 Total 48 50

Practice Solution-- Jefferson
Modified Divisor = 1,003,464 / 53 Initial Modified Divisor = 18,933 Is this the Ideal Ratio???? Modified Quotas (18,933) Modified Lower Quotas Jefferson Final Apportionments Brooke 2.58 2 Hopkins 8.51 8 Isaac 4.61 4 Haley 31.49 31 Wallace 5.81 5 Total 50

Worksheet #3 The country of Parador has six states: Azucar, Bahia, Café, Diamonte, Esmeralda, Felicidad. Their populations are respectively, 2,646,000; 6,900,000; 3,154,000; 891,000; 3,685,000; 2,988,000 Parador has Congress with 350 seats…distributed through apportionment… 20,264,000÷350 = 57,897 standard divisor So Find: the final apportionment of seats using the Hamilton and Jefferson Methods

Hamilton Method State Azucar Bahia Total 45.7 119.17 54.47 15.38 63.64
Café Diamante Esmeralda Felicidad Total Population 2,646,000 6,900,000 3,154,000 891,000 3,685,000 2,988,000 Standard Quota 45.7 119.17 54.47 15.38 63.64 51.6 Lower Quota 45 119 54 15 63 51 347 +3 Final Apportionment 46 64 52 350 Republic of Parador: Population Data by State

Jefferson Method State Azucar Bahia Total 45.7 119.17 54.47 15.38
Café Diamante Esmeralda Felicidad Total Population 2,646,000 6,900,000 3,154,000 891,000 3,685,000 2,988,000 Standard Quota 45.7 119.17 54.47 15.38 63.64 51.6 Lower Quota 45 119 54 15 63 51 347 Modified Quota Using 57,405 46.09 46 120.19 120 54.9 15.5 64.19 64 52.05 52 351 Using 57,500 46.01 120. 54.85 15.49 64.08 51.96 350 Republic of Parador: Population Data by State

Worksheet #4 Four siblings are going to inherit their father’s estate including a house, car, collection of sports memorabilia, and bank accounts totaling $35,000. Below you will find a chart showing how much each sibling values each item. Use the Estate Division Algorithm to find the final settlement for each sibling. House Car Memorabilia Sally $150,000 $8,900 $24,000 AJ $149,000 $9,000 $15,500 Johnny $165,000 $21,500 Marcus $162,500 $9,200 $25,500

Share of remaining cash
Sally AJ Johnny Marcus Total bids and cash 217,900 208,500 230,400 232,200 Fair share 54,475 52,125 57,600 58,050 Items received -- House Car Memorabilia Value of item received 165,000 34,700 Initial cash received -107,400 23,350 Share of remaining cash 3,113 Final settlement $57,588 $55,238 $60,713 $61,163

Warm-Up Use the Jefferson method to apportion 55 seats in congress for following populations of states: Province ,000 Province ,500 Province ,000 Province ,000 Province ,000 Province ,000 Province ,000 Province ,000

Webster Method Step 1.Find a number D such that when each state's modified quota (state's population / D) is rounded the conventional way (to the nearest integer), the total is the exact number of seats to be apportioned. Step 2.Apportion to each state its modified quota rounded to the nearest integer (relative to the arithmetic mean).

Webster Example A college has 6 dorm complexes on campus: A, B, C, D, E and F. Their respective populations are 1646; 6936; 154; 2091; 685; and College uses a 25-member student advisory board to create guidelines and rules for the dorm complexes. If they choose to use the Webster Method of Apportionment to assign the dorm complexes their seats on the board, how many seats would each complex be assigned?

Another Webster Example
A county is composed of four districts: A, B, C and D. Their populations are 210; 1082; 311; and 284 respectively. The county commission has 18 seats. Use the Webster Method to find the final apportionment for each District.

Practice Problem For another example, let’s consider a country with 3 states, named A, B and C. Suppose the populations of each state are as given below A--453 B--367 C--697 Suppose that this country has a house of representatives with 75 seats. What is the standard divisor? Find the Apportionment for each state using the Webster Method.

continued What is the quota for each state ?
Is this the correct number of seats? If it is we are finished. If not, what do you think we should do…..besides give up?

continued If the sum of the Quotas is NOT equal to the correct number of seats to be apportioned, then, by TRIAL and ERROR, find a number (a modified divisor) to use in the place of the standard divisor so that when the modified quota for each state is rounded based on the arithmetic mean, the sum of all the rounded quotas is the exact number of seats available.

Webster Summarized Once we find each state’s quota, we give that state an initial apportionment equal to <q>. That is, we’ll round the quota q to the integer <q> using traditional rounding techniques. At this point, if the total apportionment we’ve assigned equals the house size then we are done. However, we may have already assigned more than the available seats or there may be extra seats available that have not yet been assigned. In either case (too many seats assigned or not enough) we will determine a modified divisor that yield the required total apportionment when rounding the normal way.

Independent Practice Problem
There are 15 scholarships to be apportioned among 231 English majors, 502 History majors, and 355 Psychology majors. How would they be apportioned if you used the Jefferson Method? Try the problem using the Webster Method of Apportionment

Independent Practice Problem
There are 15 scholarships to be apportioned among 231 English majors, 502 History majors, and 355 Psychology majors. How would they be apportioned if you used the Jefferson Method? What would change if you used these methods if the number of scholarships goes from 15 to 16? Try the problem using the Webster Method of Apportionment

Worksheet #1 Total Population: 3,615,920
Standard Divisor=3,615,920/105= 34,437

Hamilton Apportionment
St. Quota Lower Quota Final Apport. Virginia 18.31 18 Massachusetts 13.80 13 +1 14 Pennsylvania 12.57 12 +1 13 North Carolina 10.27 10 New York 9.63 9 +1 Maryland 8.09 8 Connecticut 6.88 6 +1 7 South Carolina 5.99 5 +1 6 New Jersey 5.21 5 New Hampshire 4.12 4 Vermont 2.48 2 Georgia 2.06 Kentucky 1.995 1 +1 Rhode Island 1.99 Delaware 1.61 Total 97 (+8) 105

Jefferson Apportionment
Find the initial modified divisor by: Truncating the lower quotas = = 112 Use this find the modified divisor by dividing the total by this number: Initial Modified Divisor = 3,615,920/112 = 32,285

Jefferson Apportionment
Mod. Quota Virginia 19.53 19 Massachusetts 14.72 14 Pennsylvania 13.41 13 North Carolina 10.95 10 New York 10.27 Maryland 8.63 8 Connecticut 7.34 7 South Carolina 6.39 6 New Jersey 5.56 5 New Hampshire 4.39 4 Vermont 2.65 2 Georgia 2.19 Kentucky 2.13 Rhode Island 2.12 Delaware 1.72 1 Total 105

Worksheet #2 Total = 750 Standard Divisor = Total / # staff
= 750 / 30 = 25 Standard Quotas Lower Quotas Hamilton Final Algebra 7.52 7 Pre-Calc 5.68 5 6 Calculus 5.52 Adv. Calculus 2.56 2 3 Discrete Math 8.72 8 9 Total 27 30

Modified Lower Quotas (23.4)
Jefferson Method Modified Divisor is 750 / 32 = 23.4 Modified Quotas Modified Lower Quotas (23.4) Jefferson Final Apportionment Algebra 8.03 8 Pre-Calc 6.07 6 Calculus 5.90 5 Adv. Calculus 2.74 2 Discrete Math 9.32 9 Totals 30

Worksheet #3 Let’s assume each of the above problems just use simple traditional rounding to find the apportionment once you find each of the Standard Quotas. Can you predict what would happen and figure out a way to fix any problem you find?

Worksheet #4 Deandre Jackie Makayla Adam Total 640,000 637,000 652,000
619,000 Fair Share 160,000 159,250 163,000 154,750 Items Beauty Shop Necklace Condo T-Bird Items Value 190,000 55,000 230,000 75,000 Cash -30,000 104,250 -67,000 79,750 Remaining Cash 9,500

Cash Line 125, , , , , , ,750 38,000 / 4 = 9,500 each

Final Settlements???

Hill Method Calculate the Standard Divisor.
Calculate each state’s Standard Quota. Initially assign a state its Lower Quota if the fractional part of its Standard Quota is less than the Geometric Mean of the two whole numbers that the Standard Quota is immediately between (for example, is immediately between 16 and 17). Initially assign a state its Upper Quota if the fractional part of its Standard Quota is greater than or equal to the Geometric Mean of the two whole numbers that the Standard Quota is immediately between (for example, is immediately between 16 and 17). [In other words, round down or up based on the geometric mean.]

Geometric Mean The GEOMETRIC MEAN, GM, of 2 numbers a & b is defined by the Square Root of the product of the two numbers a & b. So if the Standard Quota is 5.54…..then “a and b” are “5 and 6” and the Geometric Mean is the square root of 30 or 5.48 and since 5.54 is more than 5.48, you round up.

Hill Method continued Check to see if the sum of the Quotas (Lower and/or Upper from Step 3) is equal to the correct number of seats to be apportioned. If the sum of the Quotas (Lower and/or Upper from Step 3) is equal to the correct number of seats to be apportioned, then apportion to each state the number of seats equal to its Quota (Lower or Upper from Step 3). If the sum of the Quotas (Lower and/or Upper from Step 3) is NOT equal to the correct number of seats to be apportioned, then, by trial and error, find a number, MD, called the Modified Divisor to use in place of the Standard Divisor so that when the Modified Quota, MQ, for each state (computed by dividing each State's Population by MD instead of SD) is rounded based on the geometric mean, the sum of all the rounded Modified Quotas is the exact number of seats to be apportioned. Apportion each state its Modified Rounded Quota.

Practice Problem revisited
Five counties of a state can divide its 50 legislative seats in many ways. Determine the number of representatives for each of the five counties using the Hill method. The following table gives the populations of the different counties. County/ Population: Brooke / 48,859; Hopkins / 161,135; Isaac / 87,194; Haley / 596,270; and Wallace /110,006……..

Practice Populations Standard Quotas Geometric Mean Quota
Modified Quotas Brooke 48,859 Hopkins 161,135 Isaac 87,194 Haley 596,270 Wallace 110,006

Practice The country of Parador has six states: Azucar, Bahia, Café, Diamonte, Esmeralda, Felicidad. Their populations are respectively, 1,646,000; 6,936,000; 154,000; 2,091,000; 685,000; 988,000 Parador has Congress with 250 seats…distributed through apportionment using the Webster Method

Practice Problem Five counties of a state can divide its 50 legislative seats in many ways. Determine the number of representatives for each of the five counties using the Jefferson method. The following table gives the populations of the different counties. County/ Population: Brooke / 48,859; Hopkins / 161,135; Isaac / 87,194; Haley / 596,270; and Wallace /110,006……..

Jefferson Method Reviewed
Procedure: Calculate the Standard Divisor. Calculate each state’s Standard Quota. Initially assign each state its Lower Quota. Check to see if the sum of the Lower Quotas is equal to the correct number of seats to be apportioned. If the sum of the Lower Quotas is equal to the correct number of seats to be apportioned, then apportion to each state the number of seats equal to its Lower Quota. If the sum of the Lower Quotas is NOT equal to the correct number of seats to be apportioned, then, by trial and error, find a number, MD, called the Modified Divisor to use in place of the Standard Divisor so that when the Modified Quota, MQ, for each state (computed by dividing each State's Population by MD instead of SD) is rounded DOWN, the sum of all the rounded (down) Modified Quotas is the exact number of seats to be apportioned. (Note: The MD will always be smaller than the Standard Divisor.) These rounded (down) Modified Quotas are sometimes called Modified Lower Quotas. Apportion each state its Modified Lower Quota.

Webster Method Reviewed
Calculate the Standard Divisor. Calculate each state’s Standard Quota. Initially assign a state its Lower Quota if the fractional part of its Standard Quota is less than Initially assign a state its Upper Quota if the fractional part of its Standard Quota is greater than or equal to [In other words, round down or up based on the arithmetic mean (average).] Check to see if the sum of the Quotas (Lower and/or Upper from Step 3) is equal to the correct number of seats to be apportioned. If the sum of the Quotas (Lower and/or Upper from Step 3) is equal to the correct number of seats to be apportioned, then apportion to each state the number of seats equal to its Quota (Lower or Upper from Step 3). If the sum of the Quotas (Lower and/or Upper from Step 3) is NOT equal to the correct number of seats to be apportioned, then, by trial and error, find a number,MD, called the Modified Divisor to use in place of the Standard Divisor so that when the Modified Quota, MQ, for each state (computed by dividing each State's Population by MD instead of SD) is rounded based on the arithmetic mean (average) , the sum of all the rounded Modified Quotas is the exact number of seats to be apportioned. Apportion each state its Modified Rounded Quota.

Practice Estate Division
Four heirs, Aaayyy, Beeeee, Seeeee, and Deeeee, are going to inherit an estate with a house, car, $4500 in cash, and jewelry from their aunt. The table below shows their secret bids representing how much they value each item. Find the final settlement for each heir. House Car Jewelry Aaayyyy 54,000 2,900 2,600 Beeeee 49,000 3,100 2,200 Seeeee 55,000 2,850 2,550 Deeeee 53,500 3,300

Practice A small country consists of four states. The population of State 1 is 44,800, the population of State 2 is 52,200, the population of State 3 is 49,200, and the population of State 4 is 53,800. The total number of seats in the legislature is 100. Find the seats apportioned to each state using the Webster Method. Find the seats apportioned to each state based on the Hill Method.

Practice in Pairs Standard Quota Webster Modified Quota Geometric Mean
Hill Quota State 1 44,800 State 2 52,200 State 3 49,200 State 4 53,800

Vocabulary Fair Division Divider Chooser Method Lone Divider Method Lone Chooser Method Estate Division Fair Share Apportionment Hamilton Method Jefferson Method Webster Method Hill Method Ideal Ratio Modified Divisor Standard Divisor Standard Quota Lower Quota Upper Quota Proportional Quota Rule Modified Quota Alabama Paradox New States Paradox Population Paradox Geometric Mean Arithmetic Mean

Go to quizlet.live on a computer

k12insight.com/TEAL Session11.14.16BLT

Independent Practice A small country consists of four states. The population of State 1 is 1,251, the population of State 2 is 14,749, the population of State 3 is 5,651, and the population of State 4 is 3,349. The total number of seats in the legislature is 250. Find the seats apportioned to each state using the Hamilton method.

Practice Problem States A, B, C, and D have populations of 156,000; 1,310,000; 280,000; and 254,000 respectively. There are 20 seats to apportion among them in their newly formed regional economic development council. Determine the apportionment of seats using each of the Jefferson method.

Worksheet The state of Delaware has three counties: Kent, New Castle, and Sussex. The Delaware state House of Representatives has 41 members. Delaware wants to divide this representation along county lines and the populations of the counties are as follows (from the 2010 Census): County Population Kent 162,310 New Castle 538,479 Sussex 197,145.

Divisor for Jefferson is 20,882
Stdnrd Divisor is 21,901 Stndrd Quotas Lower Hamilton Final Modified Divisor for Jefferson is 20,882 Jefferson Lower Quotas & FInal Kent 7.41 7 7.77 New Castle 24.59 24 25 25.79 Sussex 9.00 9 9.44

Webster Modified Quotas Rounded
Stdnrd Divisor is 21,901 Stndrd Quotas Webster Rounded Quotas Modified Divisor is 21,950 Modified Quotas Webster Modified Quotas Rounded Kent 7.41 7 7.39 New Castle 24.59 25 24.53 Sussex 9.00 9 8.98

Stdnrd Divisor is 21,901 Stndrd Quotas Geometric Means Rounded Using GM Hill Final Kent 7.41 7.48 7 New Castle 24.59 24.49 25 Sussex 9.00 9.48 9

Use all four methods to apportion the 75 seats of Rhode Island’s House of Representatives among its five counties. County Population Bristol 49,875 Kent ,158 Newport 82,888 Providence 626,667 Washington 126,979

Divisor for Jefferson is
Stdnrd Divisor is Stndrd Quotas Lower Hamilton Final Modified Divisor for Jefferson is Jefferson Lower Quotas & FInal Bristol Kent Newport Providence Washington

Stdnrd Divisor is Stndrd Quotas Webster Rounded Quotas Modified Divisor is Modified Quotas Webster Modified Quotas Rounded Bristol Kent Newport Providence Washington

Stndrd Quotas Geometric Means Rounded Using GM Hill Final
Stdnrd Divisor is Stndrd Quotas Geometric Means Rounded Using GM Hill Final Bristol Kent Newport Providence Washington

Divisor for Jefferson is
Stdnrd Divisor is Stndrd Quotas Lower Hamilton Final Modified Divisor for Jefferson is Jefferson Lower Quotas & FInal Bristol Kent Newport Providence Washington

Stdnrd Divisor is Stndrd Quotas Webster Rounded Quotas Modified Divisor is Modified Quotas Webster Modified Quotas Rounded Bristol Kent Newport Providence Washington

Stndrd Quotas Geometric Means Rounded Using GM Hill Final
Stdnrd Divisor is Stndrd Quotas Geometric Means Rounded Using GM Hill Final Bristol Kent Newport Providence Washington

Populations Standard Quotas Initial Quotas Modified Quotas Final Apportionment Brooke 48,859 Hopkins 161,135 Isaac 87,194 Haley 596,270 Wallace 110,006

Hamilton Method Reviewed
Procedure: Calculate the Standard Divisor. Calculate each state’s Standard Quota. Initially assign each state its Lower Quota. If there are surplus seats, give them, one at a time, to states in descending order of the fractional parts of their Standard Quota.

Unit Two Test Topics Fair Division Algorithms Divider-Chooser
Lone-Divider Estate Division Apportionment Algorithms Hamilton Jefferson Webster Hill Problems History

Fair Division Divider-Chooser Algorithm Lone Divider Algorithm
AKA “the cut-and-choose” method Lone Divider Algorithm

Estate Division Algorithm
Step 1: Submit bids for the goods in the estate Step 2: Find each heir’s “fair share” Step 3: Distribute the goods from the estate to the heirs based on their bids Step 4: Distribute the cash from the estate and any heir who is required to pay in to the estate to the heirs Step 5: Summarize the process in the Final Settlement for each heir

Apportionment Find the Standard Divisor Find the Standard Quotas
Hamilton Method Find the Standard Divisor Find the Standard Quotas Round to the Lower Quota Apportion any remaining seats in order from highest to lowest fractional pieces

Practice in Pairs The Planet of Isadora is home to four major continents, Jameliana, Timora, Julietta and Frederaria whose populations are in turn (12, 295,300; 9,325,500; 17,800,000; and 5,675,000). The entire planet came together after their fifth world war and decided to be governed by a ruling council comprised of 795 members. Using the Hamilton method of apportionment, how would the seats be apportioned?

Standard Divisor?? Standard Quotas Lower Quotas Hamilton
Populatons Standard Quotas Lower Quotas Hamilton Apportionments Jameliana 12, 295,300 Timora 9,325,500 Julietta 17,800,000 Frederaria 5,675,000

Jefferson Method Find the Standard Divisor Find the Standard Quotas Round to the Lower Quota Truncate the lower quotas by adding one and find a Modified Divisor Find the Modified Lower Quotas If the Modified Lower Quotas do not apportion all of the seats, use trial and error to continue to modify the divisor

Practice in Pairs The very last widget manufacturing company in America has five widget factories placed in Podunk, Dinkytown, Backwoods, Nowhere, and Last Chance. The company has decided to expand its workforce by 295 workers because widgets are making a huge comeback in the marketplace and they have chosen the Jefferson method of apportionment based on the production of each factory. In order, the factories produce 12,000; 13,500; 11,750; 9,900; and 10,700 widgets a month. How many workers would be apportioned to each of the factories?

12,000 13,500 11,750 9,900 10,700 Podunk Dinkytown Backwoods Nowhere
Widgets Made Podunk 12,000 Dinkytown 13,500 Backwoods 11,750 Nowhere 9,900 Last Chance 10,700

Webster Method Find the Standard Divisor Find the Standard Quotas Round using the Arithmetic Mean (traditional rounding) If the Rounded Quotas apportion all the seats, you are finished If all the seats are not apportioned, then use a modified divisor to calculate modified quotas until all seats are apportioned

Practice in Pairs The Cincinnati Reds have decided to apportion their remaining salary of $12,500,000 in increments of $125,000 (100 of them) to their top five players based on their average hits per season. Further, they have decided to use the Webster Method of Apportionment to calculate how much each should receive so they don’t go over the mandated salary cap. If the chart below shows the five players and their average hits per season, how much of the new salary would be apportioned to each player? Joey Votto 175 Todd Frazier 127 Devin Mesoraco 146 Brandon Phillips 152 Jay Bruce 115

Average # Hits Joey Votto 175 Todd Frazier 127 Devin Mesoraco 146 Brandon Phillips 152 Jay Bruce 115

Hill Method Find the Standard Divisor Find the Standard Quotas Calculate the Geometric Mean Compare the Standard Quota against the Geometric Mean and round up or down If the Rounded Quotas apportion all the seats, you are finished If all the seats are not apportioned, then use a modified divisor to calculate modified quotas and re-compare to your geometric means and round until all seats are apportioned

Practice in Pairs The local school system has over a hundred schools and, because it is so spread out, it operates five independent bus fleets in five areas of the district (East, West, Central, North and South). The superintendent has decided to apportion 50 new buses to the five areas based on their riderships each day. East buses 5,260 students, West buses 6,700 students, Central buses 9,780 students, North buses 3,500 students and South buses 8,750 students each day. The school board elects to use the Hill Method of Apportionment to determine where the new buses are assigned. How many new buses will each area receive?

Ridership East 5260 West 6700 Central 9780 North 3500 South 8750

Warm-Up Problem The Planet of Isadora is home to four major continents, Jameliana, Timora, Julietta and Frederaria whose populations are in turn (12, 295,300; 9,325,500; 17,800,000; and 5,675,000). The entire planet came together after their fifth world war and decided to be governed by a ruling council comprised of 795 members. Using the Hill method of apportionment, how would the seats be apportioned?

Warm-Up 2 The very last widget manufacturing company in America has five widget factories placed in Podunk, Dinkytown, Backwoods, Nowhere, and Last Chance. The company has decided to expand its workforce by 295 workers because widgets are making a huge comeback in the marketplace and they have chosen the Jefferson method of apportionment based on the production of each factory. In order, the factories produce 12,000; 13,500; 11,750; 9,900; and 10,700 widgets a month. How many workers would be apportioned to each of the factories?

Warm-Up Problem The Cincinnati Reds have decided to apportion their remaining salary of $12,500,000 in increments of $125,000 (100 of them) to their top five players based on their average hits per season. Further, they have decided to use the Jefferson Method of Apportionment to calculate how much each should receive so they don’t go over the mandated salary cap. If the chart below shows the five players and their average hits per season, how much of the new salary would be apportioned to each player? Joey Votto 175 Zach Cozart 127 Devin Mesoraco 146 Brandon Phillips 152 Billy Hamilton 115

Independent Practice The local school system has over a hundred schools and, because it is so spread out, it operates five independent bus fleets in five areas of the district (East, West, Central, North and South). The superintendent has decided to apportion 50 new buses to the five areas based on their riderships each day. East buses 5,260 students, West buses 6,700 students, Central buses 9,780 students, North buses 3,500 students and South buses 8,750 students each day. The school board elects to use the Jefferson and Hill Methods of Apportionment to determine where the new buses are assigned. How many new buses will each area receive?

Warm-Up Problem Five friends recently visited a Subway sandwich shop and decided to buy a three foot long party sub. Sammy contributed $4.50, Johnny put in $8.00, Alan threw in $5.50, Susie contributed $5.00 and Jim put in $ If they decide to distribute the sandwich using the Hamilton Method of Apportionment based on the amount each paid, how many inches of the sub would each person get?

Remember This Scenario
The sophomore, junior and senior classes of Cox Mill High School have 398, 337 and 319 member respectively. If the student council is composed of 20 members divided among each of the three classes, determine a fair number of seats on the council for each class…and explain why you chose that approach…and why you think it is fair. How might this problem look using Hamilton’s Algorithm?

Further Practice The student council has 20 members at Central High for the sophomore, junior, and senior classes that have 459, 244 and 197 students respectively. Using the Jefferson method, how many seats would each class be entitled to? What happens if the number of student council members is moved to 21 to avoid tie votes?

Continued At the beginning of the second semester, the enrollment has changed to 460, 274 and 196 students in each class and a re- apportionment of seats is called for by the senior class. Using the Hamilton method, re-calculate the apportionment of seats….. Do you see a problem?

Practice Problem The Banana Republic has states Apure (3,310,000); Barinas (2,670,000); Carabobo (1,330,000); and Dolores (690,000) and 160 seats in the legislature, with populations in parentheses. Use the Webster Method to apportion the seats.

Practice Problem A mathematics department has 30 teaching assistants to be divided among three courses, according to their respective enrollments. Suppose the enrollments are as follows. Apportion the teaching assistants among the three courses using the Hamilton Method. Then Re-Apportion the TAs using 31 teaching assistants. Course College Algebra Statistics Liberal Arts Math Total Enrollment 978 500 322 1800

Fair Division / Apportionment

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